New results in Chern-Simons theory

Summary

In these lectures, I will describe some aspects of an approach to the Jones polynomial of knots, and its generalizations, that is based on a three-dimensional quantum Yang-Mills theory in which the usual Yang-Mills Lagrangian is replaced by a Chern-Simons action. This approach gives a manifestly three-dimensional approach to the subject, but some of the key aspects of the story, involving the Feynman path integral, are somewhat beyond the reach of present rigorous understanding. In the first two lectures, I will describe aspects of the subject that can be developed rigorously at present. This basically consists of a gauge theory approach to the Jones representations of the braid group and their generalizations. In the last lecture, I will describe the more ambitious Feynman path integral approach, which is an essential part of the way that physicists actually think about problems such as this one, and which gives the most far-reaching results.

The first two lectures describe joint work with S. Delia Pietra and S. Axelrod. In this work, methods of symplectic geometry and canonical quantization are used to associate a vector space ℋ_Σ to every oriented surface Σ. In this approach, we will have to pick a complex structure J on Σ as an auxiliary tool in constructing the ℋ_Σ and much of the effort will go into constructing a projectively flat connection whose existence shows that in a suitable sense, the ℋ_Σ are independent of J.